FIG. 1A is a block diagram illustrating a signal model of a typical system. An input x is sent to system 100, which applies a function F to x to generate an output y. Since it is possible for the function to vary over time, F is also dependent on time t. In many systems, it is desirable for the system function F(x, t) to be linear. FIG. 1B is a three-dimensional (3-D) diagram illustrating a linear relationship between the input x, the output y, and time t. In this ideal case the input and the output have a linear relationship that is constant throughout time t. Thus, function y=F(x, t) forms a plane in the 3-D diagram.
It is common, however, for the system to be nonlinear. There are many possible causes for system nonlinearities, including characteristics of nonlinear components (such as conductors, capacitors and transistors), the input signal's frequency subrange, history and rate of change (also referred to “slew rate”), as well as external factors such as operating temperature. FIG. 1C is a 3-D diagram illustrating a typical nonlinear relationship. In this example, not only is the relationship between input x and output y nonlinear, this nonlinear relationship changes over time, as illustrated by sample functions F1(x, t1), F2(x, t2), and F3(x, t3). Thus, the function y=F(x, t) forms a nonlinear manifold. It is often useful to have filters that can implement this type of nonlinear functions. Many existing nonlinear filters, however, are complex, expensive, and unstable.